A348309 - OEIS

%I #20 Oct 18 2022 15:30:46

%S 1,1,1,1,1,1,1,1,0,-4,-14,-34,-69,-125,-209,-329,-493,-705,-955,-1199,

%T -1324,-1092,-56,2560,8025,18313,36353,66273,113525,184653,286257,

%U 422377,589028,763912,888378,837502,372835,-928725,-3776537,-9302337,-19226889,-36034869,-63099331,-104630831,-165212760

%N a(n) = Sum_{k=0..floor(n/8)} (-1)^k * binomial(n-4*k,4*k).

%H Seiichi Manyama, <a href="/A348309/b348309.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec#order_08">Index entries for linear recurrences with constant coefficients</a>, signature (4,-6,4,-1,0,0,0,-1).

%F G.f.: (1-x)^3/((1-x)^4 + x^8).

%F a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4) - a(n-8).

%t LinearRecurrence[{4, -6, 4, -1, 0, 0, 0, -1}, {1, 1, 1, 1, 1, 1, 1, 1}, 45] (* _Amiram Eldar_, Oct 11 2021 *)

%o (PARI) a(n) = sum(k=0, n\8, (-1)^k*binomial(n-4*k, 4*k));

%o (PARI) my(N=66, x='x+O('x^N)); Vec((1-x)^3/((1-x)^4+x^8))

%Y Cf. A348308, A348310.

%Y Cf. A099586, A348289.

%K sign,easy

%O 0,10

%A _Seiichi Manyama_, Oct 11 2021